Coupled charge and spin dynamics in a photoexcited doped Mott insulator

Using a nonequilibrium implementation of the extended dynamical mean-field theory (EDMFT) we simulate the relaxation after photoexcitation in a strongly correlated electron system with antiferromagnetic spin interactions. We consider the $t-J$ model and focus on the interplay between the charge and spin dynamics in different excitation and doping regimes. The appearance of string states after a weak photoexcitation manifests itself in a nontrivial scaling of the relaxation time with the exchange coupling and leads to a correlated oscillatory evolution of the kinetic energy and spin-spin correlation function. A strong excitation of the system, on the other hand, suppresses the spin correlations and results in a relaxation that is controlled by hole scattering. We discuss the possibility of detecting string states in optical and cold-atom experiments.

Figure 1

(a) Equilibrium phase diagram of the $t-J$ model in the space of temperature $T$ and doping $\delta$. (b) Local spectral function $A\left(\omega \right)$ for $J=0.3$ and different dopings ($\delta =0.01,0.05,0.15,0.20$) at temperatures $T=0.05$ (solid lines) and $T=0.2$ (dashed lines). The different line colors correspond to different dopings, as indicated in panel (a).

Figure 2

Intensity plots of the $\mathbf{k}$-dependent spectral functions $A_{\mathbf{k}}\left(\omega \right)$ at $T=0.5$ (upper panels) and at $T=0.05$ (lower panels) for $J=0.3$ and different dopings (left panels: $\delta =0.05$; right panels: $\delta =0.20$).

Figure 3

Intensity plot of the $\mathbf{k}$-dependent spin-spin correlation function $\mathrm{Im}{\chi }_{\mathbf{k}}\left(\omega \right)$ at $T=0.5$ (upper panels) and at $T=0.05$ (lower panels) for $J=0.3$ and different dopings (left panels: $\delta =0.05$; right panels: $\delta =0.20$).

Figure 4

(a) Relaxation dynamics of the normalized kinetic energy $[E_{\mathrm{kin}}\left(t\right)/E_{\mathrm{kin}}\left(0\right)]$ after a weak quench excitation $A(t<0)=0\to A(t\ge 0)=0.35$. Different lines correspond to different doping and temperature values as indicated in the phase diagram in the inset. (b) $J$ dependence of $E_{\mathrm{kin}}\left(t\right)$ with the rescaled time $t\to t{J}^{2/3}$. Results are shown for $T=0.05$ and doping $\delta =0.05$ (black lines) and $\delta =0.2$ (red lines).

Figure 5

Spin-spin correlation function at $\mathbf{k}=(\pi ,\pi )$ vs kinetic energy. Black squares represent the relation between the two observables in equilibrium at different temperatures. The blue and red lines show nonequilibrium results for $\delta =0.05$ at $T=0.05$ after a weak $\left[A\right(t<0)=0\to A(t\ge 0)=0.35]$ and strong $\left[A\right(t<0)=0\to A(t\ge 0)=1.4]$ quench excitation, respectively. The inset shows a zoom of the spiral behavior after the initial relaxation for the weak excitation.

Figure 6

(a) Time evolution of the normalized kinetic energy $[E_{\mathrm{kin}}\left(t\right)/E_{\mathrm{kin}}\left(0\right)]$ after a strong quench excitation $A(t<0)=0\to A(t\ge 0)=1.4$. The parameters of the $t-J$-model used for the different lines are the same as in Fig. 4 (the additional orange lines correspond to doping $\delta =0.10$). (b) Relaxation time ${\tau }_{\mathrm{dec}}$ vs temperature $T$ for different doping values. Error bars indicate the uncertainties in the fit with Eq. (18). (c) Relaxation time ${\tau }_{\mathrm{dec}}$ vs strength of the vector potential $A$ at fixed temperature $T=0.05$ and for different doping values.

Figure 7

Temporal evolution of the spectral function $A(t,\omega )$ (left panels), the real part of the impurity retarded spin-spin interaction $\mathrm{Re}{\mathcal{J}}^R(t,\omega )$ (middle panels), and the real part of the on-site (dashed lines) and nearest-neighbor (solid lines) screened effective lattice interaction $\mathrm{Re}{W}_{ij}^R(t,\omega )-J_{ij}$ (right panels). The dynamics of the underdoped ($\delta =0.05$) spin system with $J=0.3$ at $T=0.05$ is shown after the weak and strong excitations in panels (a1)–(c1) and (a2)–(c2), respectively, while the temporal evolution of the overdoped ($\delta =0.20$) system is presented in panels (d1)–(f1) and (d2)–(f2). The black dashed lines in the left and middle panels represent initial equilibrium results. The insets in the middle panels show the real part of the time-dependent impurity spin-spin interaction $\mathrm{Re}{\mathcal{J}}^R$ in the static limit ($\omega =0$).

Figure 8

Time-dependent optical conductivity $\sigma (\omega ,t)$ in the underdoped case ($\delta =0.05$) at $T=0.05$ after (a1)–(b1) the weak quench excitation with $A(t<0)=0\to A(t\ge 0)=0.35$ and (a2)–(b2) the strong quench excitation with $A(t<0)=0\to A(t\ge 0)=1.4$. In panels (a1) and (a2) the real (solid lines) and the imaginary (dashed lines) part of the optical conductivity are shown for the equilibrium case (black), at $t=1$ (red), and $t=36$ (blue). The inset illustrates the time-dependent height of the Drude peak $\mathrm{Re}\left[\sigma \right](\omega =0,t)$. In panels (b1) and (b2) the time-dependent change of the optical conductivity with respect to $\mathrm{Re}\left[\sigma \right](\omega ,t=1.0)$ is presented as an intensity plot. The intensity values are normalized to $\max \left(\mathrm{Re}\right[\sigma \left]\right(\omega ,t)-\mathrm{Re}[\sigma \left]\right(\omega ,t=1.0)$). Panels (c1) and (c2) plot the current density $j_{\mathrm{pr}}\left(t_{\mathrm{pr}}\right)$ induced by a probe pulse as a function of the probing delay time $t_{\mathrm{pr}}$ for the weak and strong excitation, respectively. The form of the probe pulse for $t_{\mathrm{pr}}=0$ is shown in the inset.

Figure 9

Intensity plots of the $\mathbf{k}$-dependent spectral function at constant energy $A_{\mathbf{k}}(\omega =0)$ and temperature $T=0.05$ for different doping values: (a) $\delta =0.05$, (b) $\delta =0.15$, (c) $\delta =0.20$, and (d) $\delta =0.27$. The results are plotted for the upper-right quadrant of the first Brillouin zone.

Figure 10

(a) Relaxation dynamics of the normalized kinetic energy $[E_{\mathrm{kin}}\left(t\right)/E_{\mathrm{kin}}\left(0\right)]$ after a pulse excitation with $E_0=2.0, \omega =6.0$, and ${\tau }_w=2.1$ for $J=0.3$ in the underdoped ($\delta =0.05$, black line) and overdoped ($\delta =0.20$, red line) cases at $T=0.05$. (b) $J$ dependence of $E_{\mathrm{kin}}\left(t\right)$ plotted as a function of the rescaled time $t{J}^{2/3}$. The different line colors correspond to the same dopings as in panel (a).

Figure 11

Relaxation time ${\tau }_{\mathrm{dec}}$ vs temperature $T$ for different doping values after an electric field quench with a constant excitation energy per hole ($\mathrm{\Delta }E/\delta =0.1$). Error bars indicate the uncertainties in the fitting with Eq. (18).